11,-1>, and 10,0> by guessing that they were the same linear combinations of 11/2,1/2>1 × 11/2,1/2>2, |1/2,1/2>1 × [1/2,-1/2>2, 11/2,-1/2>1 © I1/2,1/2>2, and |1/2,-1/2>1 11/2,-1/2>2 that we had previously found as eigenstates of the spin-spin Hamiltonian, and then verifying that these linear combinations satisfied equations (1) and (2) for the appropriate values of s and m. In this problem, you will find the same result in a different way. (a) Using the rule that m = m₁ + m2, find the state 11,-1> as a direct product of single-particle states. Verify by checking that your result satisfies equations (1) and (2) above. (b) By applying the raising operator to the state 11,-1> from (a), find the state 11,0> in terms of single-particle states. Verify by checking that your result satisfies equations (1) and (2) above. (c) By applying the raising operator to the state 11,0> from (b), find the state 11,1>. in terms of single-particle states Verify by checking that your result satisfies equations (1) and (2) above. (d) By using the requirements that m = m₁ + m2 and that 11,0> and 10, 0> must be orthogonal, find the state 10,0>. Verify by checking that your result satisfies equations (1) and (2) above. Recall that for a system of two particles, we can construct states Is, m> that are eigenstates of both Sz = S1z + S2z and S² = (S₁ + S2)² These states must satisfy S² Is,m> = s (s+1) ħ 2 Is,m> equation (1) Sz ls,m> = m ħ ls,m> equation (2) Recall that these states Is, m> can be found as linear combinations of the direct product states Is₁, m₁> > Is2, m2>. Consider the system of two spin ½ particles, so that s₁ = s2 = 1/2. As found in class, the two particles form a system with either spin 1 or spin 0. In class and in the textbook, we found the states 11,1>, 11,0>, 11,-1>, and 10,0> by guessing that they were the same linear combinations of 11/2, 1/2>1 × 11/2,1/2>2, |1/2,1/2>1 × |1/2,-1/2>2, 11/2,-1/2>1 1/2,1/2>2, and
11,-1>, and 10,0> by guessing that they were the same linear combinations of 11/2,1/2>1 × 11/2,1/2>2, |1/2,1/2>1 × [1/2,-1/2>2, 11/2,-1/2>1 © I1/2,1/2>2, and |1/2,-1/2>1 11/2,-1/2>2 that we had previously found as eigenstates of the spin-spin Hamiltonian, and then verifying that these linear combinations satisfied equations (1) and (2) for the appropriate values of s and m. In this problem, you will find the same result in a different way. (a) Using the rule that m = m₁ + m2, find the state 11,-1> as a direct product of single-particle states. Verify by checking that your result satisfies equations (1) and (2) above. (b) By applying the raising operator to the state 11,-1> from (a), find the state 11,0> in terms of single-particle states. Verify by checking that your result satisfies equations (1) and (2) above. (c) By applying the raising operator to the state 11,0> from (b), find the state 11,1>. in terms of single-particle states Verify by checking that your result satisfies equations (1) and (2) above. (d) By using the requirements that m = m₁ + m2 and that 11,0> and 10, 0> must be orthogonal, find the state 10,0>. Verify by checking that your result satisfies equations (1) and (2) above. Recall that for a system of two particles, we can construct states Is, m> that are eigenstates of both Sz = S1z + S2z and S² = (S₁ + S2)² These states must satisfy S² Is,m> = s (s+1) ħ 2 Is,m> equation (1) Sz ls,m> = m ħ ls,m> equation (2) Recall that these states Is, m> can be found as linear combinations of the direct product states Is₁, m₁> > Is2, m2>. Consider the system of two spin ½ particles, so that s₁ = s2 = 1/2. As found in class, the two particles form a system with either spin 1 or spin 0. In class and in the textbook, we found the states 11,1>, 11,0>, 11,-1>, and 10,0> by guessing that they were the same linear combinations of 11/2, 1/2>1 × 11/2,1/2>2, |1/2,1/2>1 × |1/2,-1/2>2, 11/2,-1/2>1 1/2,1/2>2, and
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Transcribed Image Text:11,-1>, and 10,0> by guessing that they were the same
linear combinations of 11/2,1/2>1 × 11/2,1/2>2,
|1/2,1/2>1 × [1/2,-1/2>2, 11/2,-1/2>1 © I1/2,1/2>2, and
|1/2,-1/2>1 11/2,-1/2>2 that we had previously found
as eigenstates of the spin-spin Hamiltonian, and then
verifying that these linear combinations satisfied
equations (1) and (2) for the appropriate values of s and
m.
In this problem, you will find the same result in a
different way.
(a) Using the rule that m = m₁ + m2, find the state
11,-1> as a direct product of single-particle states. Verify
by checking that your result satisfies equations (1) and
(2) above.
(b) By applying the raising operator to the state 11,-1>
from (a), find the state 11,0> in terms of single-particle
states. Verify by checking that your result satisfies
equations (1) and (2) above.
(c) By applying the raising operator to the state 11,0>
from (b), find the state 11,1>. in terms of single-particle
states Verify by checking that your result satisfies
equations (1) and (2) above.
(d) By using the requirements that m = m₁ + m2 and
that 11,0> and 10, 0> must be orthogonal, find the state
10,0>. Verify by checking that your result satisfies
equations (1) and (2) above.
Transcribed Image Text:Recall that for a system of two particles, we can
construct states Is, m> that are eigenstates of both
Sz = S1z + S2z
and
S² = (S₁ + S2)²
These states must satisfy
S² Is,m> = s (s+1) ħ 2 Is,m> equation (1)
Sz ls,m> = m ħ ls,m> equation (2)
Recall that these states Is, m> can be found as linear
combinations of the direct product states Is₁, m₁> >
Is2, m2>.
Consider the system of two spin ½ particles, so that s₁
= s2 = 1/2. As found in class, the two particles form a
system with either spin 1 or spin 0. In class and in the
textbook, we found the states 11,1>, 11,0>,
11,-1>, and 10,0> by guessing that they were the same
linear combinations of 11/2, 1/2>1 × 11/2,1/2>2,
|1/2,1/2>1 × |1/2,-1/2>2, 11/2,-1/2>1 1/2,1/2>2, and
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